
CBSE Important Questions›

CBSE Previous Year Question Papers›
 CBSE Previous Year Question Papers
 CBSE Previous Year Question Papers Class 12
 CBSE Previous Year Question Papers Class 10

CBSE Revision Notes›

CBSE Syllabus›

CBSE Extra Questions›

CBSE Sample Papers›
 CBSE Sample Papers
 CBSE Sample Question Papers For Class 5
 CBSE Sample Question Papers For Class 4
 CBSE Sample Question Papers For Class 3
 CBSE Sample Question Papers For Class 2
 CBSE Sample Question Papers For Class 1
 CBSE Sample Question Papers For Class 12
 CBSE Sample Question Papers For Class 11
 CBSE Sample Question Papers For Class 10
 CBSE Sample Question Papers For Class 9
 CBSE Sample Question Papers For Class 8
 CBSE Sample Question Papers For Class 7
 CBSE Sample Question Papers For Class 6

ISC & ICSE Syllabus›

ICSE Question Paper›
 ICSE Question Paper
 ISC Class 12 Question Paper
 ICSE Class 10 Question Paper

ICSE Sample Question Papers›
 ICSE Sample Question Papers
 ISC Sample Question Papers For Class 12
 ISC Sample Question Papers For Class 11
 ICSE Sample Question Papers For Class 10
 ICSE Sample Question Papers For Class 9
 ICSE Sample Question Papers For Class 8
 ICSE Sample Question Papers For Class 7
 ICSE Sample Question Papers For Class 6

ICSE Revision Notes›
 ICSE Revision Notes
 ICSE Class 9 Revision Notes
 ICSE Class 10 Revision Notes

ICSE Important Questions›

Maharashtra board›

RajasthanBoard›
 RajasthanBoard

Andhrapradesh Board›
 Andhrapradesh Board
 AP Board Sample Question Paper
 AP Board syllabus
 AP Board Previous Year Question Paper

Telangana Board›

Tamilnadu Board›

NCERT Solutions Class 12›
 NCERT Solutions Class 12
 NCERT Solutions Class 12 Economics
 NCERT Solutions Class 12 English
 NCERT Solutions Class 12 Hindi
 NCERT Solutions Class 12 Maths
 NCERT Solutions Class 12 Physics
 NCERT Solutions Class 12 Accountancy
 NCERT Solutions Class 12 Biology
 NCERT Solutions Class 12 Chemistry
 NCERT Solutions Class 12 Commerce

NCERT Solutions Class 10›

NCERT Solutions Class 11›
 NCERT Solutions Class 11
 NCERT Solutions Class 11 Statistics
 NCERT Solutions Class 11 Accountancy
 NCERT Solutions Class 11 Biology
 NCERT Solutions Class 11 Chemistry
 NCERT Solutions Class 11 Commerce
 NCERT Solutions Class 11 English
 NCERT Solutions Class 11 Hindi
 NCERT Solutions Class 11 Maths
 NCERT Solutions Class 11 Physics

NCERT Solutions Class 9›

NCERT Solutions Class 8›

NCERT Solutions Class 7›

NCERT Solutions Class 6›

NCERT Solutions Class 5›
 NCERT Solutions Class 5
 NCERT Solutions Class 5 EVS
 NCERT Solutions Class 5 English
 NCERT Solutions Class 5 Maths

NCERT Solutions Class 4›

NCERT Solutions Class 3›

NCERT Solutions Class 2›
 NCERT Solutions Class 2
 NCERT Solutions Class 2 Hindi
 NCERT Solutions Class 2 Maths
 NCERT Solutions Class 2 English

NCERT Solutions Class 1›
 NCERT Solutions Class 1
 NCERT Solutions Class 1 English
 NCERT Solutions Class 1 Hindi
 NCERT Solutions Class 1 Maths

JEE Main Question Papers›

JEE Main Syllabus›
 JEE Main Syllabus
 JEE Main Chemistry Syllabus
 JEE Main Maths Syllabus
 JEE Main Physics Syllabus

JEE Main Questions›
 JEE Main Questions
 JEE Main Maths Questions
 JEE Main Physics Questions
 JEE Main Chemistry Questions

JEE Main Mock Test›
 JEE Main Mock Test

JEE Main Revision Notes›
 JEE Main Revision Notes

JEE Main Sample Papers›
 JEE Main Sample Papers

JEE Advanced Question Papers›

JEE Advanced Syllabus›
 JEE Advanced Syllabus

JEE Advanced Mock Test›
 JEE Advanced Mock Test

JEE Advanced Questions›
 JEE Advanced Questions
 JEE Advanced Chemistry Questions
 JEE Advanced Maths Questions
 JEE Advanced Physics Questions

JEE Advanced Sample Papers›
 JEE Advanced Sample Papers

NEET Eligibility Criteria›
 NEET Eligibility Criteria

NEET Question Papers›

NEET Sample Papers›
 NEET Sample Papers

NEET Syllabus›

NEET Mock Test›
 NEET Mock Test

NCERT Books Class 9›
 NCERT Books Class 9

NCERT Books Class 8›
 NCERT Books Class 8

NCERT Books Class 7›
 NCERT Books Class 7

NCERT Books Class 6›
 NCERT Books Class 6

NCERT Books Class 5›
 NCERT Books Class 5

NCERT Books Class 4›
 NCERT Books Class 4

NCERT Books Class 3›
 NCERT Books Class 3

NCERT Books Class 2›
 NCERT Books Class 2

NCERT Books Class 1›
 NCERT Books Class 1

NCERT Books Class 12›
 NCERT Books Class 12

NCERT Books Class 11›
 NCERT Books Class 11

NCERT Books Class 10›
 NCERT Books Class 10

Chemistry Full Forms›
 Chemistry Full Forms

Biology Full Forms›
 Biology Full Forms

Physics Full Forms›
 Physics Full Forms

Educational Full Form›
 Educational Full Form

Examination Full Forms›
 Examination Full Forms

Algebra Formulas›
 Algebra Formulas

Chemistry Formulas›
 Chemistry Formulas

Geometry Formulas›
 Geometry Formulas

Math Formulas›
 Math Formulas

Physics Formulas›
 Physics Formulas

Trigonometry Formulas›
 Trigonometry Formulas

CUET Admit Card›
 CUET Admit Card

CUET Application Form›
 CUET Application Form

CUET Counselling›
 CUET Counselling

CUET Cutoff›
 CUET Cutoff

CUET Previous Year Question Papers›
 CUET Previous Year Question Papers

CUET Results›
 CUET Results

CUET Sample Papers›
 CUET Sample Papers

CUET Syllabus›
 CUET Syllabus

CUET Eligibility Criteria›
 CUET Eligibility Criteria

CUET Exam Centers›
 CUET Exam Centers

CUET Exam Dates›
 CUET Exam Dates

CUET Exam Pattern›
 CUET Exam Pattern
Class 10 Mathematics Revision Notes for Triangles of Chapter 6
Class 10 is one of the most crucial classes. The course is comprehensive and requires consistent practice. Stepbystep preparation allows students to score well in class 10 Mathematics. This starts with practising the questions with proper Class 10 Mathematics Chapter 6 Notes. It does become challenging for students to access and refer to all the study material, essential questions, CBSE Syllabus, etc for Class 10 Mathematics. To simplify this, Extramarks has provided Class 10 Chapter 6 Mathematics Notes which can be referred to by students. The revision notes have been prepared according to the NCERT Books.
You can also practice the CBSE previous question papers after reviewing the Class 10 Mathematics Chapter 6 Notes. This will help students to score better in class 10 examinations. You can also download the notes for future reference from the Extramarks website.
Class 10 Maths Revision Notes for Triangles of Chapter 6
Class 10 Triangles Notes – An Overview of the Chapter
Introduction
Remember Class IX where students were taught that if two figures are the same size and shape, they are said to be congruent. In this chapter, students will learn about figures that share the same shape but not necessarily the same size, they are called similar figures. Students will study the similarities between triangles in particular and use this information to provide a concise explanation of the Pythagoras Theorem.
Note All congruent figures are similar but all similar figures need not be congruent.
Similar Figures
Two polygons of the same number of sides are similar when :
(i) their corresponding angles are equal
(ii) their corresponding sides are in the same ratio (or proportion)
Similarity of Triangles
The theorem of similarity of triangles states that two triangles are similar if the two situations below are correct:
 The corresponding angles of both triangles are equal.
 The corresponding sides of the triangles are in the same proportion or ratio.
Consider that equiangular triangles are formed when the corresponding angles of two triangles are equal.
Thales, a prominent Greek mathematician, revealed some essential insight about two equiangular triangles, which is as follows:
 Any two comparable sides of two equiangular triangles have the same ratio.
He is said to have employed a result known as the Basic Proportionality Theorem.
For example:
Draw any angle XAY and on its one arm AX, mark points (say five points) P, Q, D, R and
B such that AP = PQ = QD = DR = RB.
Now, through B, draw any line intersecting the arm AY at C.
Also, through point D, draw a line parallel to BC to intersect AC at E. Observe from the constructions that ADDB = 32.
Measure AE and EC.
What about AEEC? Observe that AEEC is also equal to 32.
Thus, you can see that in ∆ ABC, DE  BC and
AD DB = AEEC
It is due to the Basic Proportionality Theorem.
Theorems in Similarity
Similarity of triangles holds three theorems. These are as follows:
Theorem 1: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
This can be proved in the following manner:
In triangle ABC, a line parallel to BC intersects AB and AC at D and E, respectively.
Theorem 2: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
This theorem can be proved by taking a line DE such that ADDB = AEEC and assuming that DE is not parallel to BC.
Theorem 3: If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion), and hence the two triangles are similar.
Two triangles are similar if they meet the AAA (Angle–Angle–Angle) criteria. Taking two triangles, ABC and DEF shows that A = D, B = E, and C = F.
Note: By the angle sum property of a triangle, two angles of one triangle will equal two angles of another triangle. It is, therefore, possible to formulate the AAA similarity criterion as follows:
A triangle is similar to another triangle if two angles of one triangle are equal to two angles of another triangle.
Theorem 4: If in two triangles, the sides of one triangle are proportional to (i.e., in the same ratio of ) the sides of the other triangle, then their corresponding angles are equal, and hence the two triangles are similar.
This criterion is the SSS (Side–Side–Side) similarity criterion for two triangles.
This theorem can be proved by taking two triangles, ABC and DEF such that ABDE = BCEF = CAFD.
SSS Criterion of Similarity
Sideside similarity criteria state that triangles with the same ratio of sides and angles will be considered similar when their corresponding sides are in the same ratio.
SAS Criterion of Similarity
According to the sideangleside similarity criteria, triangles are similar when their sides, including their angles, are in the same proportion.
Basic Proportionality Theorem
The fundamental proportionality theorem in mathematics asserts that “if a line is drawn parallel to one side of a triangle and intersects the other two sides in distinct spots, then the other two sides are split in the same ratio.”
You can prove the fundamental proportionality theorem in the following manner:
 Consider the triangle ABC. Draw a line PQ parallel to the side BC of ABC and intersect the sides AB and AC in P and Q, respectively. Thus, according to the basic proportionality theorem, AP/PB = AQ/QC.
Areas of Similar Triangles
The theorem of the area of similar triangles states that “The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
For instance: Consider two triangles – ΔABC and ΔPQR.
Area of triangle ABC/Area of Triangle QPR = (AB/PQ)^2 = (BC/QR)^2 = (CA/RP)^2.
Proof: We are given two triangles ABC and PQR, such that ∆ ABC ~ ∆ PQR.
To prove that ar(ABC) ar(PQR) = (AB\PQ)^2 = (BCQR)^2 = (CARP)^2.
We draw altitudes AM and PN of the triangles to find the areas of the two triangles.
ar (ABC) = 1/2 BC × AM and ar (PQR) = 1/2 QR × PN
So, ar (ABC)ar (PQR) = ½ BC × AM  ½ QR × PN
Now, in ∆ ABM and ∆ PQN.
∠ B = ∠ Q (As ∆ ABC ~ ∆ PQR)
∠ M = ∠ N (Each is 90°)
So, ∆ ABM ~ ∆ PQN (AA similarity criterion)
Therefore, AM/PN = AB/PQ
Also, ∆ ABC ~ ∆ PQR (Given)
So, AB/PQ = BC/QR = CA/RP
Therefore, ar(ABC)/ar(PQR) = AB/PQ * AM/PN = (AB/PQ)^2
Hence Proved.
Proof of Pythagoras Theorem Using Similarity
Pythagoras’ theorem states, “In a rightangled triangle, the sum of squares of two sides of a right triangle is equal to the square of the hypotenuse of the triangle.”
Right triangles have a hypotenuse whose square equals the sum of both sides’ squares.
The formula is: Hypotenuse^2 = Base^2 + Perpendicular^2
The proof is as follows:
Consider the rightangle triangle ABC with CB perpendicular, CA hypotenuse, and AB as a base of the triangle. Consider the right triangle, rightangled at B.
Draw BD ⊥ AC
Now, △ADB ~ △ABC
So, AD/AB = AB/AC
or AD. AC = AB2 ……………(i)
Also, △BDC ~ △ ABC
So, CD/BC = BC/AC
or CD. AC = BC2 ……………(ii)
Adding (i) and (ii),
 AC + CD. AC = AB2 + BC2
AC(AD + DC) = AB2 + BC2
AC(AC) = AB2 + BC2
⇒ AC2 = AB2 + BC2
Hence, proved.
Now Let us Use this Theorem to Prove Pythagoras’ Theorem: – repeat
The converse of Pythagoras Theorem
Statement: If the length of a triangle is a, b and c and c^2 = a^2 + b^2, then the triangle is a rightangle triangle.
Proof: Construct another triangle, △EGF, such as AC = EG = b and BC = FG = a.
Note: The formula for the rightangle triangle is as follows: a^2+b^2 = c^2.
In △EGF, by Pythagoras Theorem:
EF^2 = EG^2 + FG^2 = b^2 + a^2 …………(1)
In △ABC, by Pythagoras Theorem:
AB2 = AC2 + BC2 = b^2 + a^2 …………(2)
From equations (1) and (2), we have;
EF^2 = AB^2
EF = AB
⇒ △ ACB ≅ △EGF (By SSS postulate)
⇒ ∠G is a right angle
Thus, △EGF is a right triangle.
Hence, we can say that the converse of the Pythagorean theorem also holds.
Hence Proved.
FAQs (Frequently Asked Questions)
1. What is explicitly a triangle?
A triangle is defined as a polygon with three sides and three angles. A triangle’s interior angles add up to 180 degrees, while its exterior angles add up to 360 degrees.
2. What identically is the SSS Similarity Criterion?
According to the SSS criterion, two triangles are comparable if their respective sides are proportionate.
3. What are the main critical points from Class 10 Chapter 6 Mathematics Notes?
Following are the key points from Class 10 Chapter 6 Mathematics Notes:
 Similar figures are two figures with the same shape but not necessarily the same size.
 The congruent figures are all similar, while the opposite is not valid.
 If their corresponding angles are equal
 Their corresponding sides are in the same ratio; two polygons with the same number of sides are comparable.
 If a line is drawn parallel to one side of a triangle and intersects at different points, the other two sides are divided in the same proportion.
 A line is parallel to the third side if it divides any two sides of a triangle in the same ratio.
4. What are RHS similarity criteria?
If the hypotenuse and one side of one right triangle are proportionate to the hypotenuse and one side of the second right triangle, the two triangles are comparable. This is known as the RHS Similarity Criterion.
5. What is a Pythagoras' Theorem?
The square of the hypotenuse of a right triangle equals the sum of the squares of the other two sides (Pythagoras Theorem). If the square of one side of a triangle equals the sum of the squares of the other two sides, the angle opposite the first side is a right angle.